19.2 (2018)

For a subgroup $L$ of a group $G$, let $L_G$ denote the largest normal subgroup of $G$ contained in $L$. If $M \triangleleft G$, then we say that $G$ nearly splits over $M$ if there is a subgroup $N \leqslant G$ such that $|G : N| = \infty$, $|G : MN| < \infty$, and $(M \cap N)_G = 1$.

Let $G$ be any group, and $H$ a normal subgroup of prime order. Is it true that $\psi(G) \cap H = 1$ if and only if $G$ nearly splits over $H$?